This upper bound has the following implications:
First, it is a dimension-free bound in which \(p\)  does not appear, and this statistical dimension is described by the effective rank \(\mathbf {r}\left(\Sigma \right)\)  except for the effect of \(t\) .
This is identical to the statistical dimension of the independent case of [1]} and others.
Second, the effect of this dependence appears as \(\Gamma _{n}\)  in the coefficient part \((2\kappa ^{2}+\Gamma _{n})\)  of the upper bound.
If \(\mathbf {M}\)  is independent, we have \(\Gamma _{n} = 0\) . Then, our upper bound corresponds to the upper bound by [1]} up to the universal constants.
